American Institute of Mathematical Sciences

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May  2017, 37(5): 2881-2897. doi: 10.3934/dcds.2017124

Persistence and stationary distribution of a stochastic predator-prey model under regime switching

Li Zu 1, , Daqing Jiang 2,3,, and  Donal O'Regan 4,

1. 

College of Mathematics and Statistics, Hainan Normal University, Hainan 571158, China
2. 

School of Science, China University of Petroleum (East China), Qingdao 266580, China
3. 

Nonlinear Analysis and Applied Mathematics(NAAM)-Research Group, King Abdulaziz University, Jeddah, Saudi Arabia
4. 

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

* Corresponding author: Daqing Jiang (School of Science, China University of Petroleum (East China), Qingdao), China E-mail: daqingjiang2010@hotmail.com, Tel.: +86 43185099589; fax: +86 43185098237

Received  October 2014 Revised  January 2017 Published  February 2017

Fund Project: The first author is supported by NSFC of China (No: 11601038), the second author is supported by NSFC of China (No: 11371085) and the Fundamental Research Funds for the Central Universities (No: 15CX08011A).

Taking both white noise and colored environment noise into account, a predator-prey model is proposed. In this paper, our main aim is to study the stationary distribution of the solution and obtain the threshold between persistence in mean and the extinction of the stochastic system with regime switching. Some simulation figures are presented to support the analytical findings.

Keywords: Stochastic predator-prey model, threshold, Markov switching, stationary distribution, extinction.
Mathematics Subject Classification: Primary: 34F05, 92D25; Secondary: 92Bxx.
Citation: Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124
References:
[1]

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H. W. Hethcote, W. Wang, L. T. Han and Z. E. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268.  doi: 10.1016/j.tpb.2004.06.010.  Google Scholar

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C. Y. Ji, D. Q. Jiang and N. Z. Shi, A note on a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 377 (2011), 435-440.  doi: 10.1016/j.jmaa.2010.11.008.  Google Scholar

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H. H. Li, D. Q. Jiang, F. Z. Cong and H. X. Li, Persistence and non-persistence of a predator-prey system with stochastic perturbation Abstr. Appl. Anal. 2014 (2014), Article ID 720283, 10 pages. doi: 10.1155/2014/720283.  Google Scholar

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X. Y. Li, D. Q. Jiang and X. R. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448.  doi: 10.1016/j.cam.2009.06.021.  Google Scholar

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X. Y. Li, A. Gray, D. Q. Jiang and X. R. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28.  doi: 10.1016/j.jmaa.2010.10.053.  Google Scholar

[9]

X. Y. Li and X. R. Mao, A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica, 48 (2012), 2329-2334.  doi: 10.1016/j.automatica.2012.06.045.  Google Scholar

[10]

H. Liu, Q. S. Yang and D. Q. Jiang, The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences, Automatica, 48 (2012), 820-825.  doi: 10.1016/j.automatica.2012.02.010.  Google Scholar

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H. Liu, X. X. Li and Q. S. Yang, The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Syst. Control Lett., 62 (2013), 805-810.  doi: 10.1016/j.sysconle.2013.06.002.  Google Scholar

[12]

M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching Ⅱ, Math. Comput. Model., 55 (2012), 405-418.  doi: 10.1016/j.mcm.2011.08.019.  Google Scholar

[13]

M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Model., 54 (2011), 2139-2154.  doi: 10.1016/j.mcm.2011.05.023.  Google Scholar

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X. R. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.  doi: 10.1016/S0022-247X(03)00539-0.  Google Scholar

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X. R. Mao, Stationary distribution of stochastic population systems, Syst. Control Lett., 60 (2011), 398-405.  doi: 10.1016/j.sysconle.2011.02.013.  Google Scholar

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X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching Imperial College Press, London, 2006.  Google Scholar

[17]

R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations Wiley-Interscience, New York, 1982. Google Scholar

[18]

E. C. Pielou, Introduction to Mathematical Ecology Wiley-Interscience, New York, 1969. Google Scholar

[19]

A. Settati and A. Lahrouz, Stationary distribution of stochastic population systems under regime switching, Appl. Math. Comput., 244 (2014), 235-243.  doi: 10.1016/j.amc.2014.07.012.  Google Scholar

[20]

Y. Takeuchi, N. H. Dub, N. T. Hieu and K. Satoa, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.  doi: 10.1016/j.jmaa.2005.11.009.  Google Scholar

[21]

D. Y. Xu, B. Li, S. J. Long and L. Y. Teng, Moment estimate and existence for solutions of stochastic functional differential equations, Nonlinear Anal., 108 (2014), 128-143.  doi: 10.1016/j.na.2014.05.004.  Google Scholar

[22]

D. Y. Xu, X. H. Wang and Z. G. Yang, Further results on existence-uniqueness for stochastic functional differential equations, Sci. China Math., 56 (2013), 1169-1180.  doi: 10.1007/s11425-012-4553-1.  Google Scholar

[23]

T. Zhao, Y. Kung and H. L. Smith, Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal. Theor., 28 (1997), 1373-1394.  doi: 10.1016/0362-546X(95)00230-S.  Google Scholar

[24]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, Siam. J. Control Optim., 46 (2007), 1155-1179.  doi: 10.1137/060649343.  Google Scholar

[25]

L. Zu, D. Q. Jiang and D. O'Regan, Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regimes witching, Commun. Nonlinear Sci. Numer. Simul., 29 (2015), 1-11.  doi: 10.1016/j.cnsns.2015.04.008.  Google Scholar

show all references

References:
[1]

G. Q. Cai and Y. K. Lin, Stochastic analysis of predator-prey type ecosystems, Ecol. Complex., 4 (2007), 242-249.  doi: 10.1016/j.ecocom.2007.06.011.  Google Scholar

[2]

L. S. Chen and Z. J. Jing, The existence and uniqueness of limit cycles for the differential equations of predator-prey interactions, Chinese Sci. Bull., 9 (1984), 521-523.   Google Scholar

[3]

H. W. Hethcote, W. Wang, L. T. Han and Z. E. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268.  doi: 10.1016/j.tpb.2004.06.010.  Google Scholar

[4]

C. Y. Ji, D. Q. Jiang and N. Z. Shi, A note on a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 377 (2011), 435-440.  doi: 10.1016/j.jmaa.2010.11.008.  Google Scholar

[5]

R. Z. Khasminskii, C. Zhu and G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), 1037-1051.  doi: 10.1016/j.spa.2006.12.001.  Google Scholar

[6]

H. H. Li, D. Q. Jiang, F. Z. Cong and H. X. Li, Persistence and non-persistence of a predator-prey system with stochastic perturbation Abstr. Appl. Anal. 2014 (2014), Article ID 720283, 10 pages. doi: 10.1155/2014/720283.  Google Scholar

[7]

X. Y. Li, D. Q. Jiang and X. R. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448.  doi: 10.1016/j.cam.2009.06.021.  Google Scholar

[8]

X. Y. Li, A. Gray, D. Q. Jiang and X. R. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28.  doi: 10.1016/j.jmaa.2010.10.053.  Google Scholar

[9]

X. Y. Li and X. R. Mao, A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica, 48 (2012), 2329-2334.  doi: 10.1016/j.automatica.2012.06.045.  Google Scholar

[10]

H. Liu, Q. S. Yang and D. Q. Jiang, The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences, Automatica, 48 (2012), 820-825.  doi: 10.1016/j.automatica.2012.02.010.  Google Scholar

[11]

H. Liu, X. X. Li and Q. S. Yang, The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Syst. Control Lett., 62 (2013), 805-810.  doi: 10.1016/j.sysconle.2013.06.002.  Google Scholar

[12]

M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching Ⅱ, Math. Comput. Model., 55 (2012), 405-418.  doi: 10.1016/j.mcm.2011.08.019.  Google Scholar

[13]

M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Model., 54 (2011), 2139-2154.  doi: 10.1016/j.mcm.2011.05.023.  Google Scholar

[14]

X. R. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.  doi: 10.1016/S0022-247X(03)00539-0.  Google Scholar

[15]

X. R. Mao, Stationary distribution of stochastic population systems, Syst. Control Lett., 60 (2011), 398-405.  doi: 10.1016/j.sysconle.2011.02.013.  Google Scholar

[16]

X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching Imperial College Press, London, 2006.  Google Scholar

[17]

R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations Wiley-Interscience, New York, 1982. Google Scholar

[18]

E. C. Pielou, Introduction to Mathematical Ecology Wiley-Interscience, New York, 1969. Google Scholar

[19]

A. Settati and A. Lahrouz, Stationary distribution of stochastic population systems under regime switching, Appl. Math. Comput., 244 (2014), 235-243.  doi: 10.1016/j.amc.2014.07.012.  Google Scholar

[20]

Y. Takeuchi, N. H. Dub, N. T. Hieu and K. Satoa, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.  doi: 10.1016/j.jmaa.2005.11.009.  Google Scholar

[21]

D. Y. Xu, B. Li, S. J. Long and L. Y. Teng, Moment estimate and existence for solutions of stochastic functional differential equations, Nonlinear Anal., 108 (2014), 128-143.  doi: 10.1016/j.na.2014.05.004.  Google Scholar

[22]

D. Y. Xu, X. H. Wang and Z. G. Yang, Further results on existence-uniqueness for stochastic functional differential equations, Sci. China Math., 56 (2013), 1169-1180.  doi: 10.1007/s11425-012-4553-1.  Google Scholar

[23]

T. Zhao, Y. Kung and H. L. Smith, Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal. Theor., 28 (1997), 1373-1394.  doi: 10.1016/0362-546X(95)00230-S.  Google Scholar

[24]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, Siam. J. Control Optim., 46 (2007), 1155-1179.  doi: 10.1137/060649343.  Google Scholar

[25]

L. Zu, D. Q. Jiang and D. O'Regan, Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regimes witching, Commun. Nonlinear Sci. Numer. Simul., 29 (2015), 1-11.  doi: 10.1016/j.cnsns.2015.04.008.  Google Scholar

Figure 1.  The curves on the subgraphs (a) and (b) are the density functions of x(t) and y(t) in k = 1, k = 2, respectively. The white noise intensity on x(t) and y(t) are all relatively small.
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Figure 2.  The subgraphs (a) and (b) denote sample phase portrait of stochastic system and the corresponding deterministic system. The red, blue and black area denote x(t), y(t) in k = 1, k = 2 and converting between the above two states, respectively. The white noise intensity on x(t) and y(t) are all relatively small.
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Figure 3.  The subgraphs (a) and (b) have the same definitions as in Fig.2. The white noise intensity on x(t) and y(t) are all relatively big.
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